This 2nd edition textbook offers a rigorous introduction to measure theoretic probability with particular attention to topics of interest to mathematical statisticians-a textbook for courses in probability for students in mathematical statistics. It is recommended to anyone interested in the probability underlying modern statistics, providing a solid grounding in the probabilistic tools and techniques necessary to do theoretical research in statistics. For the teaching of probability theory to post graduate statistics students, this is one of the most attractive books available.

Of particular interest is a presentation of the major central limit theorems via Stein's method either prior to or alternative to a characteristic function presentation. Additionally, there is considerable emphasis placed on the quantile function as well as the distribution function. The bootstrap and trimming are both presented. Martingale coverage includes coverage of censored data martingales. The text includes measure theoretic preliminaries, from which the authors own course typically includes selected coverage.

This is a heavily reworked and considerably shortened version of the first edition of this textbook. "Extra" and background material has been either removed or moved to the appendices and important rearrangement of chapters has taken place to facilitate this book's intended use as a textbook.

New to this edition:

• Still up front and central in the book, Chapters 1-5 provide the "measure theory" necessary for the rest of the textbook and Chapters 6-7 adapt that measure-theoretic background to the special needs of probability theory
• Develops both mathematical tools and specialized probabilistic tools
• Chapters organized by number of lectures to cover requisite topics, optional lectures, and self-study
• Exercises interspersed within the text

• Guidance provided to instructors to help in choosing topics of emphasis

  Autorentext

  Galen Shorack, PhD, is Professor Emeritus in the Department of Statistics (of which he was a founding member) and Adjunct Professor in the Department of Mathematics at the University of Washington, Seattle. He received his Bachelor of Science and Master of Science degrees in Mathematics from the University of Oregon and his PhD in Statistics from Stanford University. Dr. Shorack's research interests include limit theorems in statistics, the theory of empirical processes, trimming-Winsorizing, and regular variation. He has served as Associate Editor of the Annals of Mathematical Statistics (Annals of Statistics) and is Fellow of the Institute of Mathematical Statistics.

  Klappentext

  The choice of examples used in this text clearly illustrate its use for a one-year graduate course. The material to be presented in the classroom constitutes a little more than half the text, while the rest of the text provides background, offers different routes that could be pursued in the classroom, as well as additional material that is appropriate for self-study. Of particular interest is a presentation of the major central limit theorems via Steins method either prior to or alternative to a characteristic function presentation. Additionally, there is considerable emphasis placed on the quantile function as well as the distribution function, with both the bootstrap and trimming presented. The section on martingales covers censored data martingales.

  Inhalt

  PrefaceUse of This TextDefinition of Symbols Chapter 1. Measures

1. Basic Properties of Measures
2. Construction and Extension of Measures

3. Lebesgue Stieltjes MeasuresChapter 2. Measurable Functions and Convergence

4. Mappings and s-Fields
5. Measurable Functions
6. Convergence
7. Probability, RVs, and Convergence in Law

8. Discussion of Sub s-FieldsChapter 3. Integration

9. The Lebesgue Integral
10. Fundamental Properties of Integrals
11. Evaluating and Differentiating Integrals
12. Inequalities

13. Modes of ConvergenceChapter 4 Derivatives via Signed Measures

14. Introduction
15. Decomposition of Signed Measures
16. The Radon Nikodym Theorem
17. Lebesgue's Theorem

18. The Fundamental Theorem of CalculusChapter 5. Measures and Processes on Products

19. Finite-Dimensional Product Spaces
20. Random Vectors on (O, ,P)
21. Countably Infinite Product Probability Spaces

22. Random Elements and Processes on (O, ,P)Chapter 6. Distribution and Quantile Functions

23. Character of Distribution Functions
24. Properties of Distribution Functions
25. The Quantile Transformation
26. Integration by Parts Applied to Moments
27. Important Statistical Quantities

28. Infinite VariancesChapter 7. Independence and Conditional Distributions

29. Independence
30. The Tail s-Field
31. Uncorrelated Random Variables
32. Basic Properties of Conditional Expectation

33. Regular Conditional ProbabilityChapter 8. WLLN, SLLN, LIL, and Series

34. Introduction
35. Borel Cantelli and Kronecker Lemmas
36. Truncation, WLLN, and Review of Inequalities
37. Maximal Inequalities and Symmetrization
38. The Classical Laws of Large Numbers (or, LLNs)
39. Applications of the Laws of Large Numbers
40. Law of the Iterated Logarithm (or, LIL)
41. Strong Markov Property for Sums of IID RVs
42. Convergence of Series of Independent RVs
43. Martinagles

44. Maximal Inequalities, Some with BoundariesChapter 9. Characteristic Functions and Determining Classes

45. Classical Convergence in Distribution
46. Determining Classes of Functions
47. Characteristic Functions, with Basic Results
48. Uniqueness and Inversion
49. The Continuity Theorem
50. Elementary Complex and Fourier Analysis
51. Esseen's Lemma
52. Distributions on Grids

53. Conditions for Ø to Be a Characteristic FunctionChapter 10. CLTs via Characteristic Functions

54. Introduction
55. Basic Limit Theorems
56. Variations on the Classical CLT
57. Examples of Limiting Distributions
58. Local Limit Theorems
59. Normality Via Winsorization and Truncation
60. Identically Distributed RVs
61. A Converse of the Classical CLT
62. Bootstrapping

63. Bootstrapping with Slowly WinsorizationChapter 11. Infinitely Divisible and Stable Distributions

64. Infinitely Divisible Distributions
65. Stable Distributions
66. Characterizing Stable Laws
67. The Domain of Attraction of a Stable Law
68. Gamma Approximations

69. Edgeworth ExpansionsChapter 12. Brownian Motion and Empirical Processes

70. Special Spaces
71. Existence of Processes on (C, C) and (D, D)
72. Brownian Motion and Brownian Bridge
73. Stopping Times
74. Strong Markov Property
75. Embedding a RV in Brownian Motion
76. Barrier Crossing Probabilities
77. Embedding the Partial Sum Process
78. Other Properties of Brownian Motion
79. Various Empirical Processes
80. Inequalities for the Various Empirical Processes

81. ApplicationsChapter 13. Martingales

82. Basic Technicalities for Martingales
83. Simple Optional Sampling Theorem
84. The Submartingale Convergence Theorem
85. Applications of the S-mg Convergence Theorem
86. Decomposition of a Submartingale Sequence
87. Optional Sampling
88. Applications of Optional Sampling
89. Introduction to Counting Process Martingales

90. CLTs for Dependent RVsChapter 14. Convergence in Law on Metric Spaces

91. Convergence in Distribution on Metric Spaces

92. Metrics for Convergence in Distribution Chapter 15. Asymptotics Via Empirical Processes

93. Introduction
94. Trimmed and Winsorized Means
95. Linear Rank Statistics and Finite Sampling

96. L-StatisticsAppendix A. Special DistributionsElementary ProbabilityDistribution Theory for StatisticsAppendix B. General Topology and Hilbert Space

97. General Topology
98. Metric Spaces

99. Hilbert SpaceAppendix C. More WLLN and CLT

100. Introduction
101. General Moment Estimation Specific
102. Slowly Varying Partial Variance when s2=8
103. Specific Tail Relationships
104. Regularly Varying Functions
105. Some Winsorized Variance Comparison
106. Inequalities for Winsorized Quantile…